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The expansion of half-integral polytopes

Jean Cardinal, Lionel Pournin

TL;DR

This paper investigates edge expansion for graphs arising from polytopes with half-integral vertices. The authors show a negative result for general half-integral polytopes by constructing $d$-dimensional instances with exponentially small expansion in $d$, while proving a positive result for half-integral zonotopes: their expansion is uniformly bounded away from zero, at least $7/12$. A key intermediate finding is that half-integral zonotopes are affine images of graphical zonotopes of cycles and hypercubes, enabling a contraction of the problem to well-controlled cycle- and cube-based graphs. The main technique combines a congestion-based analysis (Sinclair’s method) with structural zonotope decompositions, yielding sharp lower bounds on expansion and highlighting a dichotomy between general half-integral and zonotopal cases with implications for related combinatorial optimization problems.

Abstract

The expansion of a polytope is an important parameter for the analysis of the random walks on its graph. A conjecture of Mihai and Vazirani states that all $0/1$-polytopes have expansion at least 1. We show that the generalization to half-integral polytopes does not hold by constructing $d$-dimensional half-integral polytopes whose expansion decreases exponentially fast with $d$. We also prove that the expansion of half-integral zonotopes is uniformly bounded away from $0$. As an intermediate result, we show that half-integral zonotopes are always graphical.

The expansion of half-integral polytopes

TL;DR

This paper investigates edge expansion for graphs arising from polytopes with half-integral vertices. The authors show a negative result for general half-integral polytopes by constructing -dimensional instances with exponentially small expansion in , while proving a positive result for half-integral zonotopes: their expansion is uniformly bounded away from zero, at least . A key intermediate finding is that half-integral zonotopes are affine images of graphical zonotopes of cycles and hypercubes, enabling a contraction of the problem to well-controlled cycle- and cube-based graphs. The main technique combines a congestion-based analysis (Sinclair’s method) with structural zonotope decompositions, yielding sharp lower bounds on expansion and highlighting a dichotomy between general half-integral and zonotopal cases with implications for related combinatorial optimization problems.

Abstract

The expansion of a polytope is an important parameter for the analysis of the random walks on its graph. A conjecture of Mihai and Vazirani states that all -polytopes have expansion at least 1. We show that the generalization to half-integral polytopes does not hold by constructing -dimensional half-integral polytopes whose expansion decreases exponentially fast with . We also prove that the expansion of half-integral zonotopes is uniformly bounded away from . As an intermediate result, we show that half-integral zonotopes are always graphical.
Paper Structure (4 sections, 19 theorems, 9 equations, 1 figure)

This paper contains 4 sections, 19 theorems, 9 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sparse cuts in half-integral polytopes
  3. Half-integral zonotopes are graphical
  4. The expansion of half-integral zonotopes

Key Result

Theorem 1.1

There exists half-integral polytopes of arbitrarily large dimension $d$ whose expansion is less than $d/\sqrt{2}^d$.

Figures (1)

  • Figure 1: The polytope $\Xi$ when $d$ is equal to $3$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 11 more